That's kinda neat you can take out the rocker switch.
Can you show a better pic of which wires go where on the Carling toggle switch?
I don't remember the pinout by heart, but it's something like the pic below (the pic may or may not be accurate, but from memory seems right).
Do you have a multimeter and comfortable using it?
On the super old winches the controller only had 3 pins. One was 12 volt feed to controller, and the other two pins got connected to the 12 volt through the switch when activated (one for in and other for out).
The 5 pin winches added in two pins that would ground the winches solenoids, so one of those pins was connected to ground and the other pin was connected to the wire on the solenoid that should go to ground. These two pins are connected when the controller switch is activated (in or out, doesn't matter).
The lights on your new controller are to indicate thermal overheating or something like that (I really don't know about that newfangled stuff).
You can pretty easily meter everything out (winch side and controller side) to see if everything matches. Since you can remove that toggle switch in your controller, it should be quite easy to rewire to fit your winch.
I would run the winch using jumper wires first to prove that it has no issues, then rewire your controller to match.